Periodically driven cold atoms may exhibit a special type of resonant
behaviour, which is due to the gravity field. In a cloud of periodically pulsed cold atoms, falling under the action of gravity, a fraction
of the atoms are steadily accelerated away from the bulk of the atomic cloud; their
acceleration is parameter dependent, and may even be opposite
in direction to gravity. This experimentally discovered effect is called "quantum accelerator modes". It has no
counterpart in the classical dynamics of the driven atoms, yet its theoretical explanation rests on a "pseudoclassical limit",
that relates the effect to the stableperiodic orbits of a fictitious dynamical system described by a map. As a consequence, several classic items of nonlinear dynamics, such as an arithmetic organization of the modes, play a role in the description of this purely quantal phenomenon .

Contents

Kicked atoms and rotors.

Non interacting particles moving in a line, subject to an external driving in the form of infinitesimally short pulses, or kicks, periodic in time and space, are a typical system studied in Cold atom experiments in Quantum Chaos. They can be described by the time-periodic Hamiltonian :
\[
\tag{1}
\hat H(t)\;=\;\frac12\hat P^2+ k\cos(\hat X)\sum\limits_{n=-\infty}^{+\infty}\delta(t-n\tau)
\]
which classically generates the Chirikov Standard Map (Chirikov 1979). In eqn.(1) dimensionless variables are used, such that the spatial period of the kicks is \(2\pi\), momentum \(P\) is in units of the Planck's constant \((2\pi)\hbar\) divided by the spatial period of the driving force, the mass of the particle is \(1\), and \(\tau\) is the time between successive kicks. The reduced Planck constant is then $1$. Due to spatial periodicity, the Bloch theorem ensures that quasi-momentum (hereafter denoted \(\beta\)) is conserved under the evolution governed by (1), so momentum is effectively quantized, and if the particle is in a well defined quasi-momentum eigenstate, then it will effectively move like a rotor - that is, a particle constrained to move in a circle, parameterized by \(\hat X\)mod\((2\pi)\). For this reason, kicked cold atoms modeled by Hamiltonian (1) provide efficient means of experimental realizations of the celebrated Kicked Rotor (KR) model (Moore, Robinson, Bharucha, Sundaram, Raizen 1995; Raizen, Steck 2011). In this way it was possible to experimentally observe the main characteristic features of that model, namely Resonances and Dynamical Localization. A natural frequency for the free rotor is the ratio between the level separation and the Planck constant. A KR resonance occurs when this natural frequency is commensurate to the driving frequency; in that case, for special values of the quasi-momentum the rotor's energy grows quadratically in time. If the natural frequency is incommensurate to the driving frequency, dynamical localization in momentum and energy occurs due to quantum destructive interference. This is quite similar to the Anderson localization in disordered solids (Fishman 2010).

Figure 1: Experimentally measured momentum distributions in the freely falling frame for increasing number of pulses.

where \(\eta/\tau\) is the gravitational acceleration. For the Oxford experiments, the period \(\tau=2\pi\) corresponds to \(66.85 \mu\)s .

The most important experimental observation in (Oberthaler et al 1999) was what has been termed a Quantum Accelerator Mode (QAM) found for a range of kicking periods close to the KR resonant value $\tau=2\pi$. This strange effect involves a certain fraction of the atoms receiving a constant amount of momentum from each pulse so that an acceleration is produced. This is different from the gravitational acceleration and, surprisingly, may even be directed upwards. Figure 1 (taken from Oberthaler et al 1999) shows the experimental momentum distribution of a cloud of cold (\(5 \, \mu K\)) caesium atoms, in the atomic ensemble's falling frame, as the pulse number is varied when the kicking period is 60.5 \(\mu\)s. As the pulse number increases part of the distribution splits off and moves away from the main peak near zero momentum (relative to that of free falling atoms). The size and shape of the accelerated peak remain basically unchanged as pulses continue to be applied. For this particular set of data, the probability that an atom remains in the accelerator mode between consecutive kicks is over 99%. This is in marked contrast to the main peak, which becomes more diffuse as the number of pulses increases.

Pseudoclassics.

The QAM phenomenon is observed near (though not exactly at) values of \(\tau\) where the KR resonates . That means the system is in a deeply quantum regime, which excludes that QAM may have a classical origin; indeed no counterpart for them is to be found in the classical mechanics corresponding to (2). Their explanation nevertheless rests on the observation, that near a KR resonance the system behaves remarkably close to a classical dynamical system, which is unrelated to the true classical limit. This "pseudoclassical" limit is derived as follows (Fishman et al 2003). The gravitational potential in (2) breaks spatial periodicity, which is restored by a gauge transformation, that amounts to measuring momentum in a free-falling frame. The resulting Hamiltonian is:
\[\tag{3}
\hat H_g(t)=\frac12(\hat P-\frac{\eta}{\tau}t)^2+k\cos(\hat X)\sum\limits_{n=-\infty}^{+\infty}\delta(t-n\tau)
\]
This Hamiltonian is space-periodic, so quasi-momentum is conserved, and Bloch reduction from particle dynamics to rotor dynamics can be performed. If a value \(\beta\) of the quasi-momentum is fixed, then the unitary rotor evolution from immediately after a kick to immediately after the next kick can be written in the form (Fishman et al 2003):
\[\tag{4}
\hat U(t)=\exp\left(-ik\cos(\theta)\right)\exp\left(-i\frac{\tau}2(\hat N+\beta-\eta\tau t-\eta/2)^2\right)
\]
where \(t\) is now an integer time variable, that counts the kicks; moreover, \(\theta=X\)mod\((2\pi)\), and \(\hat N=-i\tfrac d{d\theta}\). This evolution is different from that of the quantum KR, due to the reshuffling of phases due to gravity. Indeed dynamical localization is removed and diffusive momentum growth is observed at long-times. At short times, however, ballistic growth may occur. This is precisely due to the QAM effect and is observed near the quantum KR resonances. These occur when \(\tau=2\pi l/m\) with \(l,m\) integers. The main resonances correspond to \(m=1\). Near a main resonance let \(\tau=2\pi l+\epsilon\) with \(|\epsilon|\ll 1\); and define \(\tilde k=k|\epsilon|\), \(\hat I=\hat N\epsilon\). Then
\[
\tag{5}
\hat U(t)=\exp\left(-\frac{i}{|\epsilon|}\tilde k\cos(\theta)\right)\exp\left(-\frac{i}{|\epsilon|}\cal H_{\beta}(\hat I,t)\right)\;,
\]
where
\[\tag{6}
\cal H_{\beta}(\hat I,t)=\frac12\; \operatorname{sign}(\epsilon)\hat I^2+\hat I\left(\pi l+\tau\beta-t\eta\tau-\tau\eta/2\right)\;.
\]
In eq.(5), \(|\epsilon|\) plays the formal role of a Planck's constant, and (5) can be seen as the formal quantization of a classical map in the (pseudo)-classical canonical variables \(\theta\) and \(I\), using \(|\epsilon|\) as the Planck constant. This pseudo-classical map is time-dependent, because so is
\({\cal H}_{\beta}\). To remove time dependence one can implement a time-dependent change of variables from \(I\) to \(J=I\pm\pi l\pm \tau(\beta-\eta t-\eta/2)\), finally leading to the map:
\[
\tag{7}
J_{t+1}=J_t+\tilde k\sin(\theta_{t+1})\pm 2\pi\Omega\;,\\
\theta_{t+1}=\theta_t\pm J_t\;.
\]

where the sign \(\pm\) has to be chosen according to the sign of \(\epsilon\) and \(\Omega=\eta\tau/2\pi\) is the gravity acceleration, when space and time are measured in units of the spatial period of the kicks and of the kicking period respectively.

As the difference between \(J\) and \(I\) linearly grows in time, fixed points of map (7) give rise to linear increase of the momentum \(I\) in time , hence of the physical momentum as well. More generally , if map (7)
is read on the 2-torus, then any fixed point with period \(\mathfrak p\) and jumping index \(\mathfrak j\) of the toral map (the jumping index is defined by \(J_{t+{\mathfrak p}}=J_t+2\pi{\mathfrak j}\)) produces an acceleration in physical momentum, given by \(2\pi(\pm\Omega+{\mathfrak j}/{\mathfrak p})/|\epsilon|\). Each such fixed point is surrounded by a stable island; an example is shown in Figure 2, where \({\mathfrak j}=0\) and \({\mathfrak p}=1\). The whole island travels with the same acceleration, and if an initial wave packet overlaps such a pseudoclassical island, then part of it will be trapped a long time inside the same island, thereby producing an observable quantum accelerator mode. It was verified that such a mode decays in time due to dynamical tunneling with the effective Planck's constant \(|\epsilon|\).
The QAMs originally observed in Oxford correspond to period-1 fixed points. However, the theory predicts many more QAMs, associated with fixed points of period \({\mathfrak p}>1\). Such higher-order modes require longer observation times, and were indeed observed in subsequent experiments (Schlunk, d'Arcy, Gardiner, Summy 2003).

The pseudoclassical theory predicts that a particle with initial momentum \(P_0\), caught in a pseudoclassical \(({\mathfrak p},{\mathfrak j})\) accelerating island, after \(t\) kicks will have momentum \(
P\simeq P_0+\frac{2\pi t}{|\epsilon|}\left[\frac{\mathfrak j}{\mathfrak p}\pm\Omega\right]\)
(d'Arcy et al 2004). At fixed \(t\) this equation defines a curve in the $(\tau, P)$ plot, where the population is enhanced by the \(({\mathfrak p},{\mathfrak j})\)-mode. Such curves corresponding to different \(({\mathfrak p},{\mathfrak j})\) are compared to experimental distributions in Figure 3 and the agreement is excellent.

Figure 3: Experimentally measured momentum distributions after (a) 60 kicks and (b) 90 kicks, for different kicking periods near \(\tau=4\pi\). Darker zones correspond to higher population. The kicking period is given in \(\mu\)secs on the horizontal axis. Various accelerator modes corresponding to different \(({\mathfrak p},{\mathfrak j})\) are observed . Dotted lines mark the pseudoclassical predictions.

It can be shown that the fractions \(({\mathfrak j}/{\mathfrak p})\) which are constructed using the jumping indices \(\mathfrak j\) and the periods \(\mathfrak p\) of the pseudoclassical accelerator modes which are exhibited by map (7) are
the Farey rational approximants(Hardy and Wright 1979) to the dimensionless gravity \(\Omega\) (Buchleitner, d'Arcy, Fishman, Gardiner, Guarneri, Ma, Rebuzzini, Summy 2006) . In the plane of the parameters \(\tilde k,\Omega\) a phase diagram can be constructed, that shows which modes are observable in which regions. This diagram displays structures, that are reminiscent of the
Arnold Tongues . Indeed,
for every coprime \(({\mathfrak p},{\mathfrak j})\), there is a region in the \(\tilde k,\Omega\) plane wherein map (7)
has fixed points of period \({\mathfrak p}\) and jumping index \({\mathfrak j}\). Near the \(\tilde k=0\) axis, this region has the shape of a wedge with its tip at \(\tilde k=0\), \(\Omega= {\mathfrak j}/{\mathfrak p}\) (Hihinashvili, Oliker, Avizrats, Iomin, Fishman, Guarneri 2007; Guarneri et al 2006). At variance with conventional Arnol'd tongues, regions of existence of different modes do overlap for sufficiently large values of \(\tilde k\).

An extension of the theory (Guarneri, Rebuzzini 2008) shows that QAM also exist
near higher KR resonances, where \(\tau=2\pi l/m+\epsilon\), \(|\epsilon|\ll 1\), and \(m>1\). Such modes have been experimentally observed for \(m=3\) (Ramareddy, Behinarein, Talukdar, Ahmadi, Summy 2010).

Summary

Quantum accelerator modes are observed on kicked cold atoms in the presence of gravity. In spite of being a purely quantal phenomenon, they are explained
by a pseudoclassical limit, whereby they are associated with rational approximants to the number that represents the gravity acceleration in appropriate units.